The Jaynes–Cummings model (sometimes abbreviated JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity (or a bosonic field), with or without the presence of light (in the form of a bath of electromagnetic radiation that can cause spontaneous emission and absorption). It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity. The Jaynes–Cummings model is of great interest to atomic physics, quantum optics, solid-state physics and quantum information circuits, both experimentally and theoretically. It also has applications in coherent control and quantum information processing. Historical development 1963: Edwin Jaynes & Fred Cummings The model was originally developed in a 1963 article by Edwin Jaynes and Fred Cummings to elucidate the effects of giving a fully quantum mechanical treatment to the behavior of atoms interacting with an electromagnetic field. In order to simplify the math and allow for a tractable calculation, Jaynes and Cummings restricted their attention to the interaction of an atom with a single mode of quantum electromagnetic field. (See below for further mathematical details.) This approach is in contrast to the earlier semi-classical method, in which only the dynamics of the atom are treated quantum mechanically, while the field with which it interacts is assumed to behave according to classical electromagnetic theory. The quantum mechanical treatment of the field in the Jaynes–Cummings model reveals a number of novel features, including: The existence of Rabi oscillations between the states of the two-level system as it interacts with the quantum field. This was originally believed to be a purely quantum mechanical effect, although a semi-classical explanation for it was later provided in terms of linear dispersion and absorption A ladder of quantized energy levels, called the Jaynes-Cummings ladder, that scales in energy non-linearly as where is the total number of quanta in the coupled system. This quantization of energies and non-linear scaling is purely quantum mechanical in nature. The collapse and subsequent revivals of the probability to detect the two-level system in a given state when the field is initially in a coherent state. While the collapse has a simple classical explanation, the revivals can only be explained by the discreteness of the energy spectrum due to quantum nature of the field. To